mathematics

Hartogs's extension theorem

Hartogs's extension theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hartogs's extension theorem rather than just read about it. In short: In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction.

Key takeaways

Hartogs's extension theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hartogs's extension theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hartogs's extension theorem from memory before moving on to harder problems.

Reference excerpt

In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of n > 1 complex variables. A first version of this theorem was proved by Friedrich Hartogs, and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier Soviet literature, it is also called the Osgood–Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood. This property of holomorphic functions of several variables is also called Hartogs's phenomenon: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs-type theorems.

Historical note The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables. Today, usual proofs rely on either the Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous Cauchy–Riemann equations with compact support. The latter approach is due to Leon Ehrenpreis who initiated it in the paper (Ehrenpreis 1961). Yet another very simple proof of this result was given by Gaetano Fichera in the paper (Fichera 1957), by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function: later he extended the theorem to a certain class of partial differential operators in the paper (Fichera 1983), and his ideas were later further explored by Giuliano Bratti. Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko. Their approach is to use Ehrenpreis's fundamental principle.

Hartogs's phenomenon For example, in two variables, consider the interior domain

H ε = { z = ( z 1 , z 2 ) ∈ Δ 2 : | z 1 | < ε or 1 − ε < | z 2 | } {\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2}:|z_{1}|<\varepsilon \ \ {\text{or}}\ \ 1-\varepsilon <|z_{2}|\}}

in the two-dimensional polydisk Δ 2 = { z ∈ C 2 ; | z 1 | < 1 , | z 2 | < 1 } {\displaystyle \Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}} where 0 < ε < 1. {\displaystyle 0<\varepsilon <1.}

Theorem Hartogs (1906): Any holomorphic function f {\displaystyle f} on H ε {\displaystyle H_{\varepsilon }} can be analytically continued to Δ 2 . {\displaystyle \Delta ^{2}.} Namely, there is a holomorphic function F {\displaystyle F} on Δ 2 {\displaystyle \Delta ^{2}} such that F = f {\displaystyle F=f} on H ε . {\displaystyle H_{\varepsilon }.}

Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the domain of holomorphy.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hartogs's extension theorem

Start with the simplest possible case. Write down what Hartogs's extension theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hartogs's extension theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hartogs's extension theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hartogs's extension theorem

In research: Hartogs's extension theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hartogs's extension theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hartogs's extension theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Several complex variables, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Hartogs's extension theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hartogs's extension theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hartogs's extension theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hartogs's extension theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hartogs's extension theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hartogs's extension theorem in simple terms?

Why does Hartogs's extension theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hartogs's extension theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hartogs's extension theorem.